Riesz–Thorin interpolation theorem
E746578
The Riesz–Thorin interpolation theorem is a fundamental result in functional analysis that provides bounds for linear operators between Lᵖ spaces by interpolating their behavior between two known endpoint estimates.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Riesz–Thorin interpolation theorem canonical | 2 |
| Riesz convexity theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8640755 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Riesz–Thorin interpolation theorem Context triple: [Frigyes Riesz, knownFor, Riesz–Thorin interpolation theorem]
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A.
Riesz–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
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B.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
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C.
Bourgain–Tzafriri restricted invertibility principle
The Bourgain–Tzafriri restricted invertibility principle is a fundamental result in functional analysis and operator theory that guarantees the existence of large, well-invertible submatrices within certain classes of linear operators.
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D.
Banach–Stone theorem
The Banach–Stone theorem is a fundamental result in functional analysis that characterizes compact Hausdorff spaces via isometric isomorphisms between their spaces of continuous real- or complex-valued functions.
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E.
Gagliardo–Nirenberg interpolation inequalities
The Gagliardo–Nirenberg interpolation inequalities are fundamental results in functional analysis and partial differential equations that bound intermediate norms of functions by combinations of lower and higher order norms, playing a key role in regularity theory and nonlinear analysis.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Riesz–Thorin interpolation theorem Target entity description: The Riesz–Thorin interpolation theorem is a fundamental result in functional analysis that provides bounds for linear operators between Lᵖ spaces by interpolating their behavior between two known endpoint estimates.
-
A.
Riesz–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
-
B.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
-
C.
Bourgain–Tzafriri restricted invertibility principle
The Bourgain–Tzafriri restricted invertibility principle is a fundamental result in functional analysis and operator theory that guarantees the existence of large, well-invertible submatrices within certain classes of linear operators.
-
D.
Banach–Stone theorem
The Banach–Stone theorem is a fundamental result in functional analysis that characterizes compact Hausdorff spaces via isometric isomorphisms between their spaces of continuous real- or complex-valued functions.
-
E.
Gagliardo–Nirenberg interpolation inequalities
The Gagliardo–Nirenberg interpolation inequalities are fundamental results in functional analysis and partial differential equations that bound intermediate norms of functions by combinations of lower and higher order norms, playing a key role in regularity theory and nonlinear analysis.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
interpolation theorem
ⓘ
mathematical theorem ⓘ |
| appearsIn |
graduate textbooks on functional analysis
ⓘ
graduate textbooks on harmonic analysis ⓘ |
| applicableWhen | operator is linear ⓘ |
| appliesTo | linear operators between Lp and Lq spaces ⓘ |
| assumes | operator is bounded on two endpoint Lp spaces ⓘ |
| characteristic | interpolates exponents linearly in 1/p and 1/q ⓘ |
| concerns |
Lp spaces
ⓘ
bounded linear operators ⓘ complex interpolation ⓘ norm estimates ⓘ |
| consequence | convexity of log of operator norm in interpolation parameter ⓘ |
| contrastWith | real interpolation methods ⓘ |
| field |
functional analysis
ⓘ
harmonic analysis ⓘ operator theory ⓘ |
| generalizes |
Riesz convexity theorem
ⓘ
linked to:
Riesz–Thorin interpolation theorem
|
| gives | bounds for operator norms between Lp spaces ⓘ |
| hasVersion |
finite measure space version
ⓘ
sigma-finite measure space version ⓘ |
| historicalNote | proved independently by Marcel Riesz and Gunnar Thorin ⓘ |
| implies |
Lp boundedness from Lp0 and Lp1 bounds
ⓘ
intermediate operator norm estimate is log-convex in 1/p and 1/q ⓘ operator is bounded on intermediate Lp spaces ⓘ |
| influenced | development of modern interpolation theory ⓘ |
| involves |
holomorphic families of operators
ⓘ
strip in the complex plane ⓘ |
| isPartOf | interpolation theory of operators ⓘ |
| namedAfter |
Gunnar Thorin
ⓘ
Marcel Riesz ⓘ |
| notApplicableTo | nonlinear operators in its standard form ⓘ |
| relatedConcept |
Banach space interpolation
ⓘ
linked to:
Calderón interpolation theorem
Lp interpolation scale ⓘ |
| relatedTo |
Marcinkiewicz interpolation theorem
ⓘ
Stein interpolation theorem ⓘ
linked to:
Calderón interpolation theorem
|
| requires | measure spaces to be sigma-finite in standard formulations ⓘ |
| typeOf | complex method interpolation result ⓘ |
| typicalAssumption | operator acts on simple functions and extends by density ⓘ |
| usedIn |
Fourier analysis
ⓘ
Lp-boundedness of Fourier transform related operators ⓘ ergodic theory ⓘ estimates for convolution operators ⓘ partial differential equations ⓘ probability theory ⓘ study of singular integral operators ⓘ |
| uses |
Hadamard three-lines theorem
ⓘ
linked to:
Hadamard three-circle theorem
complex analytic methods ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Riesz–Thorin interpolation theorem Description of subject: The Riesz–Thorin interpolation theorem is a fundamental result in functional analysis that provides bounds for linear operators between Lᵖ spaces by interpolating their behavior between two known endpoint estimates.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Riesz–Thorin interpolation theorem